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%I #19 Aug 30 2018 22:44:56
%S 0,0,9,4,1,3,9,5,0,2,3,2,4,9,3,0,8,9,7,3,5,1,7,1,9,5,5,3,6,2,3,3,3,0,
%T 2,8,9,8,1,5,8,3,1,7,3,7,9,6,6,5,4,3,0,0,3,7,1,1,4,2,3,4,0,2,8,0,2,1,
%U 6,1,8,7,3,0,0,0,8,4,5,1,3,3,5,8,7,3,0,9,0,6,2,2,8,1,1,7,2,7,5,4,5,4
%N Decimal expansion of the sum of the alternating series tau(3), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 168.
%H G. C. Greubel, <a href="/A242611/b242611.txt">Table of n, a(n) for n = 0..10000</a>
%F tau(n) = -log(2)^(n+1)/(n+1) + Sum_(k=0..n-1) (binomial(n, k)*log(2)^(n-k)*gamma(k)).
%F tau(3) = gamma*log(2)^3 - (1/4)*log(2)^4 + 3*log(2)^2*gamma(1) + 3*log(2)*gamma(2).
%e 0.009413950232493089735171955362333...
%t tau[n_] := -Log[2]^(n+1)/(n+1) + Sum[Binomial[n, k]*Log[2]^(n-k)*StieltjesGamma[k], {k, 0, n-1}]; Join[{0,0},RealDigits[tau[3], 10, 100] // First]
%o (PARI) -suminf(k=1,(-1)^k*log(k)^n/k) \\ _Charles R Greathouse IV_, Mar 10 2016
%Y Cf. A001620, A082633, A086279, A242494, A242612, A242613.
%K nonn,cons
%O 0,3
%A _Jean-François Alcover_, May 19 2014
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